Computing the Approximation Error for Neural Networks with Weights Varying on Fixed Directions

Abstract

Abstract We obtain a sharp lower bound estimate for the approximation error of a continuous function by single hidden layer neural networks with a continuous activation function and weights varying on two fixed directions. We show that for a certain class of activation functions this lower bound estimate turns into equality. The obtained result provides us with a method for direct computation of the approximation error. As an application, we give a formula, which can be used to compute instantly the approximation error for a class of functions having second order partial derivatives.